The simplest system giving rise to anharmonic localized modes is a lattice with positive quartic anharmonicity (K4 > 0) in the interatomic potential
V(x) = K2/2 x2 + K4/4 x4 .

For a monatomic chain, excitations with two different parities are found: one is an odd parity mode[1] (in the electric dipole sense) that is centered at a lattice site, while the other is an even parity mode[2] with inversion symmetry with respect to the midpoint between lattice sites. The eigenvectors for these two modes are shown in the Figure.

For both modes, the neighboring particles of the chain vibrate out-of-phase with respect to each other, as would be expected for a mode which originates from an extended mode at the top of the plane wave vibrational band.

[1] A.J.Sievers and S.Takeno, Phys. Rev. Lett. 61, 970 (1988).
[2] J. B. Page, Phys. Rev. B 41, 7835 (1990).

Last modified: August 12, 1997